K-1 THE ACCELERATION DUE TO GRAVITY

(1) To study uniformly accelerated motion in the case of a freely falling body.

(2) To test the hypothesis that the acceleration of a freely falling body is constant.

(3) To find the value of that constant, called the "acceleration due to gravity," g.

2. BACKGROUND

Read any textbook's treatment of kinematics and uniformly accelerated motion.

3. THE APPARATUS

A steel plummet falls in a special apparatus which records its position as a function of
time without disturbing its motion (Figs. 2 and 3). It falls along a 1.5 meter path
between two parallel wires having a high electrical potential between them. A waxed
paper tape lies along one of the wires. The potential of the wires alternates at a
frequency of 60 Hz (derived from the building power lines). Sparks jump from one wire,
to the plummet, through the waxed tape to the other wire, leaving a mark on the tape
to mark the plummet's position. A spark occurs when the voltage is at or near its peak
value, either positive or negative, so the spark rate is 120 Hz.

The clearance between the wires and the plummet must be small. The wires must
therefore be exactly vertical to ensure that the plummet does not touch them on the way
down. This adjustment is made with the leveling screws on the three feet of the
apparatus and tested with a plumb bob.

The instructor will adjust the spark gap, and instruct you in the proper use
of the apparatus, to avoid shocks. The voltage is high, but the current is low.
Touching the wires won't kill you, but it doesn't feel good.

The plummet is initially held at the top of the apparatus by an electromagnet. The spark
source is turned on first, then the magnet power is turned off. This releases the
plummet, which falls and is caught in a felt lined (or sand filled) cup at the bottom. The
spark source is then turned off before removing the waxed tape.

4. INTERPRETATION OF DATA

Fig. 2. The spark record on tape.

The waxed paper tape with the spark marks is the basis for all subsequent analysis. The
spark marks at the beginning of the motion may be erratic and unreliable, so do not
include them in the analysis. We do not need to know where the "zero" position was,
for the analysis method uses only the changes in speed during the fall, that is,
accelerations.

Lay the data tape full length on the lab table, and tape it down at its ends (be careful
not to stretch it.) Place a two-meter stick edge down on
the tape so the stick markings lie as close to the spark
marks as possible. Without moving the stick, record the
position of every mark with respect to the stick. This is
your data.

Write your name and your partner's name on the
data tape. Carefully roll up the tape and secure
it with a paper clip. Save it in case you should
have to recheck your data. Do not hand it in with
your report unless your instructor specifically
requests it.

The data analysis methods described below use the
following procedure: The marks on the tape represent
position as a function of time. Compute the velocities
during each time interval. If the acceleration is constant
we expect these velocities to be a linear function of
time. Therefore we have two tasks:

1. To determine whether the velocity-time relation
is linear.

2. If it is linear, then compute the acceleration of the plummet.

While we usually encourage students to be creative and innovative in their approach to
data collection and analysis, this experiment, simple as it seems, has some mathematical
traps for the unwary. Even some laboratory manual authors have made the error of
recommending invalid methods of data analysis. Therefore we urge the student to follow
the following methods exactly as described.

5. METHOD 1. [All students will do this.]

The time interval between marks is 1/120 second. Since you will be making many
calculations involving time intervals you can save yourself calculation drudgery by
designating 1/120 second as the time "interval", the natural time "unit" for your selected
points. The conversion from this unit to the more proper unit "second" can be made at
the very end, when you calculate the final answers.

The numbers in this example show effects
of indeterminate experimental errors.

The first column, "mark number" may
also be interpreted as elapsed time, in the
unit "interval."

The second data column has the positions
of the spark marks. The velocities in the
third column are computed average
velocities in each "interval" of time. The
velocities are obtained by subtracting
successive entries in the position
column.

Plot the average velocities (third column)
against mark number (first column). This
plot shows the change in velocity as a
function of elapsed time. The velocities
are "average" velocities during a time
interval. As you will discover, the
velocity-time relation is linear, so the
average velocity occurs at the midpoint
of the interval, and you can plot each
point at the appropriate midpoint.

Is this plot a straight line? Should it be? If it is, find its slope in cm/interval2.

The slope of this line represents the acceleration, in the units (cm/interval)/interval =
cm/(interval2) which is usually written cm-interval2.

Choose two well-separated points on the ruler-drawn line. Use the line between these
points as the hypotenuse of a right triangle whose legs are in the direction of the
coordinate axes of the graph. The slope of the line is the quotient of the "lengths" of the
legs of this triangle. [The "lengths" are not ruler-measured lengths, but are measured
against the units of each graph axis. Therefore one "length" will be in the units
"cm/interval," and the other will have the units "interval". These are time
intervals.

Convert this value of slope to the conventional units, cm/sec2.

Since the unit "interval" is 1/120 sec, the conversion is done in this manner:

[1]

This average represents your final, "best" determination of the acceleration due to gravity,
g. Is this answer reasonable?

What do you conclude, from these graphs, about the motion of the falling body?

The worksheet and summary of results which follow are for your convenience in
laboratory. You will not hand in these pages, but will recopy and perhaps restructure
this information in your report.

6. METHOD 2. [Instructor's option.]

The analysis method of section 5 made explicit use of a velocity-time graph. A strictly
algebraic analysis is also possible. This avoids any additional error due to the errors in
making the graph.

You should still include a velocity-time graph in your report to demonstrate that the
acceleration really is constant. If it weren't, all of these methods for calculating the
acceleration would be meaningless.

If the acceleration of the plummet is constant, it has the same value whatever time
interval you choose, no matter whether the interval is large or small. The fractional
error in each calculated acceleration is smaller if you choose larger intervals.

The best data analysis methods avoid combining or comparing calculations derived from
the same data values. If the calculations use non-independent data values, subtle
"cancellation" effects can occur. This method avoids that pitfall. [See the discussion
of "successive differences" and "the method of differences" in section 8.2 of
An Introduction to Experimental Analysis by Donald E. Simanek.]

[In the following paragraphs, "interval" refers to the number in column one of the data
table, the intervals between your selected points.]

Since you have over forty calculated velocities, you can select independent pairs of them
for acceleration calculations. You might choose interval no. 1 and 21, 2 and 22, 3 and
23 etc. spaced 20 time intervals apart. For example, the acceleration calculated using
intervals 1 and 21 is

[2]

(2.3 cm/int - 0.5 cm/int)/(20 int) = 0.09 cm/int2

Do this for at least 20 pairs of intervals, to get 20 independent determinations of the
acceleration. Average these accelerations. Convert the average to the units cm/sec2 by Eq. (2) above.

7. METHOD 3. [Instructor's option.]

Group the position data into 16 or more equal time intervals
Δt.

Fig. 3. Data point grouping for method three.

Calculate velocities over the intervals labeled 1 through 8. These will have 2
Δt in the
denominator, for the time interval is 2Δt. These velocities are clearly independent, for
no two velocity calculations used the same position data points.

Now calculate four accelerations, labeled A through D in the diagram. They will all
have 4Δt in the denominator. These accelerations are clearly independent. Average them.
No position data has been used twice, so there is no possibility of data cancellation in
the computation. From the point of view of error analysis, this method has the advantage
of roughly equal weighting over the whole span of the data.

8. DISCUSSION OF RESULTS

(1) On the basis of your data, calculations and graphs can you say that the acceleration
of the plummet was constant? Within what experimental error?

(2) Considering the "scatter" of the individual acceleration values, and the scatter of the points on the graphs, what error estimate (in %) would you give for your value of g?

(3) Compare your determination of g with that in the CRC handbook. Look up
Helmert's equation in the handbook. You'll need to know that LHU is at 41° 8' North latitude, and the parking lot behind the physics
labs is 580 feet above sea level. [You can check this data by consulting a U. S. Coast
and Geodetic Survey map of the Lock Haven area.] This data, used in Helmert's
equation, gives you the "accepted" value of g with which to compare yours. What is the
percent discrepancy between your value and the accepted value?

9. QUESTIONS

(1) Suppose students at the University of Alaska did this experiment with the same
apparatus. Would you expect them to obtain the same numeric result that you did? Be
specific in explaining why or why not. What about students in the class of 2025 at Lunar
Tech, doing this experiment on the moon with the same apparatus you used?

(2) Have you demonstrated the constancy of the acceleration due to gravity? No, you
haven't, for it has different values in different places on earth! Have you demonstrated
its constancy over time at this particular location? No, you demonstrated constancy only
over a rather short time interval and within a certain range of uncertainty. You haven't
demonstrated that it will have that value tomorrow! What have you demonstrated, in this
experiment? Keep in mind that one should not claim more than is justified by the
experiment.

(3) In this experiment the plummet was released from rest. A clever student wants to
eliminate the closely spaced and unreliable sparks at the beginning of the tape. This
student designs a plummet release mechanism which gives the plummet a downward
push, only during a short interval before the beginning of the spark record. This makes
the plummet velocity larger at the beginning of the spark tape. The student's lab partner
worries that this might affect the experimental value of the acceleration due to gravity.
What effect would it have on the calculated values of velocity? What effect would it
have on the calculated value of g?

APPENDIX: USING A COMPUTER SPREADSHEET PROGRAM FOR ANALYSIS OF THE DATA

Any computer spreadsheet program is capable of performing the computations required in this
experiment. We will illustrate the method with the Multiplan &tm; spreadsheet.

Spreadsheets structure data in rows and columns, just like the data sheets you make in lab. Each entry is called a "cell". The rows and columns are labeled, usually the columns have letter
designations, the rows have number designations. We might speak of the cell C-5, which is the
cell in the third column across the sheet and the fifth row down.

A cell may contain text or values. Cells with values may contain data or results. "Value" cells may contain mathematical formulae which specify how to calculate the value which will appear in the cell.

The spreadsheet you will use has the formulae already in place. On the screen it may look
something like this:

You will notice that this looks a lot like the data sheet you would make, though it shows far more decimal places than are significant.

This particular spreadsheet does not have the capability to show some math symbols. So it uses standard symbols used in computer programming:

^ raise to a power
* multiply

Column 1 contains a series of numbers labeling the sparks.

Column 2 contains the positions of each spark. These two columns contain all of the data; all
other columns are the results of calculations on this data.

Column 3 contains the differences between entries in column 2. The formula in each cell of
column 3 is the same: it takes the number in the same row, one column to the left, subtracts the
number one column left and one row up, and enters the result.
These are the "differences" y = dL/dx. (These are really ΔL and Δx, but this spreadsheet does not allow Greek letter symbols.)

The computer performs a linear regression analysis on the numbers in column 3. This gives the
slope of the straight line which best fits the data. The process uses the squares of the y values, so a column is devoted to them. The products x*y are required also, and have their own
column.

To enter your data you must first "blank" (erase) the data already in column 2. Do this by entering B (for "blank"). Put the cursor at the first entry of column 2 and enter a colon ":".
Now move the cursor to the last data entry of column 2. (The "page down" key helps here.) The
entries to be blanked are highlighted. When this is correct, press the "enter" key. Not only are
the desired entries blanked, but any results calculated from them are also! Never fear, the
necessary formulas are still intact, and will go to work as soon as you begin to enter data in
column 2.

Do not delete any columns of this spreadsheet. You may wish to delete or add rows in the data area. But do not delete the rows labeled "First data line", "Last data line", or "Reference," since these contain formula references.

You should never use a tool without knowing how it works. The formulae being used here are
the standard ones for linear regression where the errors are predominantly in one variable
(y).

Let the data be yi and xi where i = 1, 2, ... n. We want to fit a straight line Y = mx + b to this data. (Upper case Y is used here, because values of Yi obtained from the formula for the fitted curve will not in general be the same as the data points yi). Then the slope (m) of the line and its y-intercept (b) are given by

[3,4]

The standard deviation in the slope (sm) and the standard deviationn of the y intercept (sb) are:

[5,6]

The standard deviation of the y intercept is

[7,8]

Where sy is the standard deviation of the individual data values from the fitted line. yi represents the deviation of yi from the fitted line and n is the number of data points.